nLab equivariant homotopy theory



Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts


Representation theory



Equivariant homotopy theory is homotopy theory for the case that a group GG acts on all the topological spaces or other objects involved, hence the homotopy theory of topological G-spaces.

The canonical homomorphisms of topological GG-spaces are GG-equivariant continuous functions, and the canonical choice of homotopies between these are GG-equivariant continuous homotopies (for trivial GG-action on the interval). A GG-equivariant version of the Whitehead theorem says that on G-CW complexes these GG-equivariant homotopy equivalences are equivalently those maps that induce weak homotopy equivalences on all fixed point spaces for all subgroups of GG (compact subgroups, if GG is allowed to be a Lie group). By Elmendorf's theorem, this, in turn, is equivalent to the (∞,1)-presheaves over the orbit category of GG. See below at In topological spaces – Homotopy theory.

(Beware that GG-homotopy theory is crucially different from (namely “finer” and “more geometric” than) the homotopy theory of the ∞-actions of the underlying homotopy type ∞-group of GG, and this is so even when GG is a discrete group, see below).

The union of GG-equivariant homotopy theories as GG is allowed to vary is global equivariant homotopy theory.

The direct stabilization of equivariant homotopy theory is the theory of spectra with G-action. More generally there is a concept of G-spectra and they are the subject of equivariant stable homotopy theory.

The concept of cohomology of equivariant homotopy theory is equivariant cohomology:

cohomology in the presence of ∞-group GG ∞-action:

Borel equivariant cohomologyAAAAAA\phantom{AAA}\leftarrow\phantom{AAA}general (Bredon) equivariant cohomologyAAAAAA\phantom{AAA}\rightarrow\phantom{AAA}non-equivariant cohomology with homotopy fixed point coefficients
AAH(X G,A)AA\phantom{AA}\mathbf{H}(X_G, A)\phantom{AA}trivial action on coefficients AAAA[X,A] GAA\phantom{AA}[X,A]^G\phantom{AA}trivial action on domain space XXAAH(X,A G)AA\phantom{AA}\mathbf{H}(X, A^G)\phantom{AA}

In topological spaces

Let GG be a discrete group.



A topological G-space is a topological space equipped with a GG-action.

Let I=I = \mathbb{R} be the interval object (*0I1*)({*} \stackrel{0}{\to} I \stackrel{1}{\leftarrow} {*}) regarded as a GG-space by equipping it with the trivial GG-action.


A GG-homotopy η\eta between GG-maps, f,g:XYf, g : X \to Y, is a left homotopy with respect to this II

X×*=X Id×0 f X×I η Y 1 g X×*=X. \array{ X \times {*} = X \\ {}^{\mathllap{Id \times 0}}\downarrow & \searrow^{f} \\ X \times I &\stackrel{\eta}{\to}& Y \\ {}^{\mathllap{1}}\uparrow & \nearrow_{g} \\ X\times {*} = X } \,.

(e.g. May 96, p. 15)

Homotopy theory of GG-spaces


(models for GG-equivariant spaces)

Consider the following three homotopical categories that model GG-spaces:

  1. Write

    GTop cofGTop G Top_{cof} \subset G Top

    for the full subcategory of G-CW-complexes, regarded as equipped with the structure of a category with weak equivalences by taking the weak equivalences to be the GG- homotopy equivalences according to def. .

  2. Write

    GTop loc G Top_{loc}

    for all of GTopG Top equipped with weak equivalences given by those morphisms (f:XY)GTop(f : X \to Y) \in G Top that induce for all subgroups HGH \subset G weak homotopy equivalences f H:X HY Hf^H : X^H \to Y^H on the HH-fixed point spaces, in the standard Quillen model structure on topological spaces (i.e. inducing isomorphism on homotopy groups).

  3. Write

    [Orb G op,Top loc] proj [Orb_G^{op}, Top_{loc}]_{proj}

    for the projective global model structure on functors from the opposite category of the orbit category O GO_G of GG to the Top (with its classical model structure on topological spaces).

The following theorem (the equivariant Whitehead theorem together with Elmendorf's theorem) says that these models all present the same homotopy theory.


(Elmendorf’s theorem)

The homotopy categories of all three homotopical categories in def. are equivalent:

Ho(GTop cof)Ho(GTop loc)Ho([Orb G op,Top]), Ho(G Top_{cof}) \overset{\simeq}{\longrightarrow} Ho(G Top_{loc}) \overset{\simeq}{\longrightarrow} Ho([Orb_G^{op}, Top]) \,,

where the equivalence is induced by the functor that sends a GG-space to the presheaf that it represents.

The first of these equivalences is the equivariant Whitehead theorem, the second is Elmendorf's theorem.

This is stated as (May 96, theorem VI.6.3).

Ho(GTop cof) Ho(Top cof) equivariantWhitehead Whitehead Ho(GTop loc) Ho(Top loc) Elmendorf = Ho(PSh(Orb G,Top loc)) proj Ho(*,Top loc) proj \array{ Ho(G Top_{cof}) &\underset{}{\longrightarrow}& Ho(Top_{cof}) \\ {\mathllap{\text{equivariant} \atop \text{Whitehead}}}\big\downarrow{\mathrlap{\simeq}} && {\mathllap{\simeq}}\big\downarrow{\mathrlap{\text{Whitehead}}} \\ Ho(G Top_{loc}) &\overset{}{\longrightarrow}& Ho(Top_{loc}) \\ {\mathllap{Elmendorf}}\big\downarrow{\mathrlap{\simeq}} && \downarrow^{\mathrlap{=}} \\ Ho( PSh( Orb_G, Top_{loc} ) )_{proj} &\longrightarrow& Ho( \ast, Top_{loc} )_{proj} }

(,1)(\infty,1)-category of GG-equivariant spaces

At topological ∞-groupoid it is discussed that the category Top of topological spaces may be understood as the localization of an (∞,1)-category Sh (,1)(Top)Sh_{(\infty,1)}(Top) of (∞,1)-sheaves on TopTop, at the collection of morphisms of the form {X×IX}\{X \times I \to X\} with II the real line.

The analogous statement is true for GG-spaces: the equivariant homotopy category is the homotopy localization of the category of \infty-stacks on GTopG Top.

More in detail: let GTopG Top be the site whose objects are GG-spaces that admit GG-equivariant open covers, morphisms are GG-equivariant maps and morphism YXY \to X is in the coverage if it admits a GG-equivariant splitting over such GG-equivariant open covers.


sSh(GTop) loc sSh(G Top)_{loc}

for the corresponding hypercomplete local model structure on simplicial sheaves.

Let II be the unit interval, the standard interval object in Top, equipped with the trivial GG-action, regarded as an object of GTopG Top and hence in sSh(GTop)sSh(G Top).


sSh(GTop) loc IsSh(GTop) loc sSh(G Top)_{loc}^I \stackrel{\leftarrow}{\to} sSh(G Top)_{loc}

for the left Bousfield localization at thecollection of morphisms {XId×0X×I}\{X \stackrel{Id \times 0}{\to} X \times I\}.

Then the homotopy category of sSh(GTop) loc IsSh(G Top)_{loc}^I is the equivariant homotopy category described above

Ho(sSh(GTop) loc I)GTop loc. Ho(sSh(G Top)_{loc}^{I}) \simeq G Top_{loc} \,.

This is (Morel-Voevodsky 03, example 3, p. 50).

Global equivariant homotopy theory

The above constructions may be unified to apply “for all groups at once”, this is the content of global equivariant homotopy theory.

See at cohesion of global- over G-equivariant homotopy theory.

In more general model categories

Let GG be a finite group as above. We describe the generalizaton of the above story as Top is replaced by a more general model category CC (Guillou 2006).

Definition and proposition
  1. Let CC be a cofibrantly generated model category with generating cofibrations II and generating acyclic cofibrations JJ.

    There is a cofibrantly generated model category

    [O G op,C] loc [O_G^{op}, C]_{loc}

    on the functor category from the orbit category of GG to CC by taking the generating cofibrations to be

    I O G:={G/H×i} iI,HG I_{O_G} := \{G/H \times i\}_{i \in I, H \subset G}

    and the generating acyclic cofibrations to be

    J O G:={G/H×j} jI,HG. J_{O_G} := \{G/H \times j\}_{j \in I, H \subset G} \,.
  2. Let BG\mathbf{B}G be the delooping groupoid of GG and let

    [BG op,C] loc [\mathbf{B}G^{op}, C]_{loc}

    be the functor category from BG\mathbf{B}G to CC – the category of objects in CC equipped with a GG-action equipped with a set of generating (acyclic) cofibrations

    I BG:={G/H×i} iI,HG I_{\mathbf{B}G} := \{G/H \times i\}_{i \in I, H \subset G}

    and the generating acyclic cofibrations to be

    J BG:={G/H×j} jI,HG. J_{\mathbf{B}G} := \{G/H \times j\}_{j \in I, H \subset G} \,.

    This defines a cofibrantly generated model category if [BG op,C][\mathbf{B}G^{op}, C] has a cellular fixed point functor (see…).

Definition and proposition

(generalized Elmendorf’s theorem)

There is a Quillen adjunction

G/e×():C[BG op,C] loc:() e G/e \times (-) : C \stackrel{\leftarrow}{\to} [\mathbf{B}G^{op},C]_{loc} : (-)^e

and a Quillen equivalence

Θ:[O G op,C] loc[BG op,C] loc:Φ. \Theta : [O_G^{op}, C]_{loc} \stackrel{\leftarrow}{\to} [\mathbf{B}G^{op},C]_{loc} : \Phi \,.

This is Guillou 2006, Prop. 3.1.5.

In \infty-stack (,1)(\infty,1)-toposes

The assumption on the model category CC entering the generalized Elmendorf theorem above is satisfied in particular by every left Bousfield localization

C:=L ASPSh(D) C := L_A SPSh(D)

of the global projective model structure on simplicial presheaves onany small category CC at any set AA of morphisms, i.e. for every combinatorial model category CC.

This is Guillou 2006, Ex. 4.4.

For A={C({U i})X}A = \{C(\{U_i\}) \to X\} the collection of Cech covers for all covering families of a Grothendieck topology on DD, this are the standard models for ∞-stack (∞,1)-toposes H\mathbf{H}.

This way the above theorem provides a model for GG-equivariant refinements of ∞-stack (∞,1)-toposes.


Basic facts

Elmendorf’s theorem

By Elmendorf's theorem the GG-equivariant homotopy theory is an (∞,1)-topos.

By (Rezk 14) GTopG Top is also the base (∞,1)-topos of the cohesion of the global equivariant homotopy theory sliced over BG\mathbf{B}G. See at cohesion of global- over G-equivariant homotopy theory.

Equivariant Hopf degree theorem

See at equivariant Hopf degree theorem.


The stabilization of the (∞,1)-topos GTopPSh (Orb G)G Top \simeq PSh_\infty(Orb_G) is the equivariant stable homotopy theory of spectra with G-action (“naive G-spectra”).


S 1S^1-Equivariance

circle group-equivariant homotopy theory may be presented by cyclic sets.

Equivariant homotopy theory is to equivariant stable homotopy theory as homotopy theory is to stable homotopy theory.

Rezk-global equivariant homotopy theory:

cohesive (∞,1)-toposits (∞,1)-sitebase (∞,1)-toposits (∞,1)-site
global equivariant homotopy theory PSh (Glo)PSh_\infty(Glo)global equivariant indexing category GloGlo∞Grpd PSh (*) \simeq PSh_\infty(\ast)point
sliced over terminal orbispace: PSh (Glo) /𝒩PSh_\infty(Glo)_{/\mathcal{N}}Glo /𝒩Glo_{/\mathcal{N}}orbispaces PSh (Orb)PSh_\infty(Orb)global orbit category
sliced over BG\mathbf{B}G: PSh (Glo) /BGPSh_\infty(Glo)_{/\mathbf{B}G}Glo /BGGlo_{/\mathbf{B}G}GG-equivariant homotopy theory of G-spaces L weGTopPSh (Orb G)L_{we} G Top \simeq PSh_\infty(Orb_G)GG-orbit category Orb /BG=Orb GOrb_{/\mathbf{B}G} = Orb_G


Lecture notes:

Textbooks and other accounts

The generalization of the homotopy theory of GG-spaces and of Elmendorf's theorem to that of GG-objects in more general model categories is in

and further discussed in

See also

  • Stefan Waner, Equivariant Homotopy Theory and Milnor’s Theorem, Transactions of the American Mathematical Society Vol. 258, No. 2 (Apr., 1980), pp. 351-368 (JSTOR)

Specifically with an eye towards equivariant differential topology (such as Pontryagin-Thom construction for equivariant cohomotopy):

  • Arthur Wasserman, Equivariant differential topology, Topology Vol. 8, pp. 127-150, 1969 (pdf)

Discussion in the context of global equivariant homotopy theory is in

Discussion via homotopy type theory is in

An alternative model category-structure:

  • Mehmet Akif Erdal, Aslı Güçlükan İlhan, A model structure via orbit spaces for equivariant homotopy (arXiv:1903.03152)

Last revised on November 17, 2022 at 05:02:57. See the history of this page for a list of all contributions to it.